Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,21)(13,22)(14,17)(15,20)(18,19)(23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(1,2,7,8)(3,4,6,5)(9,10,14,17)(11,15,20,16)(12,21,22,13)(18,23,24,19)(26,31)(32,35)(33,34)(39,40);(1,3,7,6)(2,5,8,4)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,4,7,5)(2,3,8,6)(9,12)(10,13)(11,18)(14,22)(15,19)(16,23)(17,21)(20,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,39)(34,40);(1,5,7,4)(2,3,8,6)(9,11)(10,16)(12,18)(13,23)(14,20)(15,17)(19,21)(22,24)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,6,7,3)(2,5,8,4)(9,13)(10,12)(11,19)(14,21)(15,18)(16,24)(17,22)(20,23)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9685663);